Mixed Discontinuous Galerkin Methods for Darcy Flow

We consider a family of mixed finite element discretizations of the Darcy flow equations using totally discontinuous elements (both for the pressure and the flux variable). Instead of using a jump stabilization as it is usually done for DG methods (see e.g. [3], [13] and the references therein) we use the stabilization introduced in [18], [17]. We show that such stabilization works for discontinuous elements as well, provided both the pressure and the flux are approximated by local polynomials of degree ≥ 1, without any need for additional jump terms. Surprisingly enough, after the elimination of the flux variable, the stabilization of [18], [17] turns out to be in some cases a sort of jump stabilization itself, and in other cases a stable combination of two originally unstable DG methods (namely, Bassi-Rebay [4] and Baumann-Oden [6]).